Non-Gaussian Merton-Black-Scholes Theory

This book introduces an analytically tractable and computationally effective class of non-Gaussian models for shocks (regular Lévy processes of the exponential type) and related analytical methods similar to the initial Merton-Black-Scholes approach, which the authors call the Merton-Black-Scholes theory.The authors have chosen applications interesting for financial engineers and specialists in financial economics, real options, and partial differential equations (especially pseudodifferential operators); specialists in stochastic processes will benefit from the use of the pseudodifferential operators technique in non-Gaussian situations. The authors also consider discrete time analogues of perpetual American options and the problem of the optimal choice of capital, and outline several possible directions in which the methods of the book can be developed further.Taking account of a diverse audience, the book has been written in such a way that it is simple at the beginning and more technical in further chapters, so that it is accessible to graduate students in relevant areas and mathematicians without prior knowledge of finance or economics.

Levy processes; regular Levy processes of the exponential type; pricing and hedging of contingent claims of the European type; perpetual American options; American options - finite time horizon; first-touch digitals; barrier options; multi-asset contracts; investment under uncertainty and capital accumulation; endogenous default and pricing of the corporate debt; fast pricing of European options; discrete time models; Feller processes of the normal inverse Gaussian type; pseudodifferential operators with constant symbols; elements of calculus of pseudodifferential operators.